How to make the insets on either side of this cyllinder symmetrical
Well, you could build the symmetrical elements first using the Mirror Modifier first.



Then apply the modifier, and work on the asymmetrical elements.



Edit

Ok, based on that additional info, here's how you could go about doing so:

Create the first cylinder with the cuts. Then duplicate it, and flip the second one so that the insets are facing each other. Then, just grab the extruded faces, and move them in/out of the cylinder face by the same distance that you did for the first one. You can type in distances after activating the grab tool for exact distances.